376 



Mr. J. C. Maxwell on Governon 



[Mar. 5, 



The equation of motion of the machine itself is 



M5^=P-E-F(|-V,)-Gy . (10) 



This must be combined with equation (7) to determine the motion of the 

 whole apparatus. The solution is of the form 



cc=k^e''ht-\-K^e'>ht-\-^^fHt-\-\ty . . . . . . (11) 



where w^, n^, are the roots of the cubic equation 



MBn' + (MY + FB)7^^ + FYn + FG= (12) 



If w be a pair of roots of this equation of the form «+ ^/ ■— lb, then the 

 part of X corresponding to these roots will be of the form 



cos (bt + ft). 



If « is a negative quantity, this will indicate an oscillation the amplitude 

 of which contiimally decreases. If a is zero, the amplitude will remain 

 constant, and if a is positive, the amplitude will continually increase. 



One root of the equation (12) is evidently a real negative quantity. 

 The condition that the real part of the other roots should be negative is 



Im"^ By B^B"^ positive quantity. 



This is the condition of stability of the motion. If it is not fulfilled 

 there will be a dancing motion of the governor, which will increase till it 

 is as great as the limits of motion of the governor. To ensure this stability, 

 the value of Y must be made sufficiently great, as compared with G, by 

 placing the weight W in a viscous liquid if the viscosity of the lubri- 

 cating materials at the axle is not sufficient. 



To determine the value of F, put the break out of gear, and fix the 

 moveable wheel ; then, if V and V be the velocities when the driving-power 

 is P and F, 



P— P' 



v-y 



To determine G, let the governor act, and let ?/ and y' be the positions 

 of the break when the driving-power is P and P', then 



P-P' 



G: 



General Theory of Chronometrie Centrifugal Pieces. 



Sir JF. Thomson''s and M. FoucauWs Governors. — Let A be the mo- 

 ment of inertia of a revolving apparatus, and d the angle of revolution. 

 The equation of motion is 



0) 



where L is the moment of the appHed force round the axis. 



